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Abstract: We study quantum graphs \(\Gamma\) with a finite or countable set \(\mathcal{E}\) of edges equipped with the Dirac operators $$\mathfrak{D}_{\Gamma,Q}=J\frac{d}{dx}+Q,J=\left( \begin{array} [c]{cc} 0 & -1\\ 1 & 0 \end{array} \right) ,$$ where $$\begin{aligned} \, Q(x) & =\left( \begin{array} [c]{cc} p(x)+r(x) & q(x)\\ q(x) & -p(x)+r(x) \end{array} \right) ,\qquad p,q,r \in L^{\infty}(\Gamma). \end{aligned}$$ We consider the self-adjointness of the unbounded operator \(\ \mathcal{D} _{\Gamma,Q,\mathfrak{M}}\) in \(L^{2}(\Gamma,\mathbb{C}^{2})\) generated by the Dirac operators \(\mathfrak{D}_{\Gamma;Q}\) with domains consisting of spinors $$u(x)=\left( \begin{array} [c]{c} u^{1}(x)\\ u^{2}(x) \end{array} \right) \in H^{1}(\Gamma,\mathbb{C}^{2}).$$ and with the coupling conditions on the vertices \(v\in\mathcal{V}\) , $$\mathfrak{M}u(v)=\mathfrak{a}_{1}(v)u^{1}(v)+\mathfrak{a}_{2}(v)u^{2} (v)=0,v\in\mathcal{V}. \qquad\qquad\qquad\qquad (1)$$ Applying the method of limit operators, we describe the essential spectra of operators \(\mathcal{D}_{\Gamma,Q,\mathfrak{M}}\) on the graphs with finite sets of exits to infinity and also on periodic graphs with aperiodic potentials and aperiodic coupling conditions. PubDate: 2022-09-01

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Abstract: It is proved that, for a sufficiently small defect of a (not necessarily bounded) quasirepresentations of an amenable group in a reflexive Banach space with dense set of bounded orbits, there is an extension of this quasirepresentation for which there is a close ordinary representation of the group in the space of this extension. PubDate: 2022-09-01

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Abstract: The paper focuses on bifurcations that occur in the essential spectrum of certain non-Hermitian operators. We consider an eigenvalue problem for an elliptic differential operator in a multidimensional tube-like domain which is infinite along one dimension and can be bounded or unbounded in other dimensions. This self-adjoint eigenvalue problem is perturbed by a small hole cut out of the domain. The boundary of the hole is described by a non-Hermitian Robin-type boundary condition. Our main result consists in sufficient conditions ensuring that this singular and non-Hermitian perturbation results in discrete eigenvalues or resonances bifurcating either from the edge or from certain internal points of the essential spectrum of the unperturbed problem. The location of the bifurcating eigenvalues and resonances is described in terms of asymptotic expansions with respect to the small linear size of the hole. Several illustrative examples are discussed. PubDate: 2022-09-01

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Abstract: Let \(T=T(h)\) be a real-analytic at \(0\in \mathbb{R} \) function, \(T(0)=\pi\) . Let \(H(x,y) = x^2+y^2 + {\cal O} _4(x,y)\) be a real-analytic at \(0\in \mathbb{R} ^2\) even function. We prove that there exists a real-analytic \( \varphi = \varphi (x) = {\cal O} _4(x)\) even function such that any solution of the Hamiltonian equations $$\dot x = \partial_y(H + \varphi ), \quad \dot y = - \partial_x(H + \varphi )$$ near the origin on the energy level \(h>0\) has the period \(T(h)\) . We discuss motivations and possible generalizations of this result. PubDate: 2022-09-01

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Abstract: For a generalized Korteweg–de Vries–Burgers equation with a flux function having two inflection points, we construct an example in which three different monotone structures of nonclassical (special) discontinuities, or undercompressive shocks, propagate at the same velocity. The linear stability of these structures is analyzed by applying the Evans function method. It is shown that some of them are neither linearly nor globally stable. The evolution of monotone unstable structures of undercompressive shocks is studied. The asymptotic behavior of the Evans function for large increments of perturbations is examined when the dissipation coefficient depends on the coordinate and time. A general approach to the construction of asymptotic expressions for the Evans function is described. PubDate: 2022-09-01

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Abstract: Dowling constructed the Dowling lattice \(Q_{n}(G)\) , for any finite set with \(n\) elements and any finite multiplicative group \(G\) of order \(m\) , which is a finite geometric lattice. He also defined the Whitney numbers of the first and second kind for any finite geometric lattice. These numbers for the Dowling lattice \(Q_{n}(G)\) are the Whitney numbers of the first kind \(V_{m}(n,k)\) and those of the second kind \(W_{m}(n,k)\) , which are given by Stirling number-like relations. In this paper, by ‘degenerating’ such relations we introduce the degenerate Whitney numbers of the first kind and those of the second kind and investigate, among other things, generating functions, recurrence relations and various explicit expressions for them. As further generalizations of the degenerate Whitney numbers of both kinds, we also consider the degenerate \(r\) -Whitney numbers of both kinds. PubDate: 2022-09-01

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Abstract: The present paper is concerned with problems of solarity, proximinality, approximative compactness, stability, and monotone path-connectedness in generalized rational approximation in spaces \(L^p\) and \(C(Q)\) . Efficiency of the new approaches and concepts used in the paper is illustrated by a number of examples. Solar properties of sets of generalized rational functions in the space \(C(Q)\) are proved with the help of the new concept of \( \mathring B \) -complete sets: a closed set \(M\) is called \( \mathring B \) -complete if, for each \(x\in X\) and \(r>0\) , the condition \(M_0:= \mathring B (x,r)\cap M)\ne\varnothing\) implies that \( {}\kern.2em\overline{\kern-.2em M}{} _0 \supset M\cap B(x,r)\) . Existence and generalized approximative compactness in problems of generalized rational approximations in spaces \(C(Q)\) and \(L^p\) are proved using the new definition of algebraic completeness with the machinery of regular Deutsch convergence. PubDate: 2022-09-01

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Abstract: This work deals with the spectral properties of the functional-difference equations that arise in a number of applications in the diffraction of waves and quantum scattering. Their link with some of the spectral properties of perturbations of the Mehler operator is addressed. The latter naturally arise in studies of functional-difference equations of the second order with a meromorphic potential which depend on a characteristic parameter. In particular, this kind of equations is frequently encountered with in the asymptotic treatment of eigenfunctions of the Robin Laplacians in wedge- or cone-shaped domains. The unperturbed selfadjoint Mehler operator is studied by means of the modified Mehler–Fock transform. Its resolvent and spectral measure are described. These results are obtained by use of some additional analysis applied to the known Mehler formulas. For a class of compact perturbations of this operator, sufficient conditions of existence and finiteness of the discrete spectrum are then discussed. Applications to the functional-difference equations are also addressed. An example of a problem leading to the study of the spectral properties for a functional-difference equation is considered. The corresponding eigenfunctions and characteristic values are found explicitly in this case. PubDate: 2022-09-01

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Abstract: We prove a continuity criterion for locally bounded finite-dimensional representations of simply connected solvable Lie groups. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020066

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Abstract: We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior of \( \sigma \) -compact subsets and, relying on this classification, we study properties of the corresponding \(C^*\) -algebras in terms of frame theory and the theory of \( {\mathscr A} \) -compact operators in Hilbert \(C^*\) -modules; some pathological examples are constructed. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020029

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Abstract: In this paper, order estimates for the Kolmogorov widths of sets defined by restrictions on the norm in the Sobolev weighted space \(W^r_{p_1}\) and the weighted space \(L_{p_0}\) are obtained. PubDate: 2022-06-01 DOI: 10.1134/S106192082202008X

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Abstract: The paper is concerned with singularities of classical solutions to the eikonal equation. With this aim in view, we study the relation between the geometry of hypersurfaces and the set of singular points of its distance function from both sides of this hypersurface. We also give an algorithm for the construction of reflection caustics and compare the results and illustrations of mathematical simulation with real-world images. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020078

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Abstract: The objective of our paper is to generalize the representation obtained in 2017 by S.Yu. Dobrokhotov, V.E. Nazaikinskii, and A.I. Shafarevich to the general case of the canonical operator on an isotropic manifold with complex germ. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020030

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Abstract: In the paper, a boundary value problem for a singularly perturbed reaction-diffusion-advection equation is considered in a two-dimensional domain in the case of discontinuous coefficients of reaction and advection, whose discontinuity occurs on a predetermined curve lying in the domain. It is shown that this problem has a solution with a sharp internal transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion in a small parameter is constructed, and also sufficient conditions are obtained for the input data of the problem under which the solution exists. The proof of the existence theorem is based on the asymptotic method of differential inequalities. It is also shown that a solution of this kind is Lyapunov asymptotically stable and locally unique. The results of the paper can be used to create mathematical models of physical phenomena at the interface between two media with different characteristics, as well as for the development of numerical-analytical methods for solving singularly perturbed problems. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020042

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Abstract: Differential-difference operators are considered on an infinite cylinder. The objective of the paper is to present an index formula for the operators in question. We define the operator symbol as a triple consisting of an internal symbol and conormal symbols on plus and minus infinity. The conormal symbols are families of operators with a parameter and periodic coefficients. Our index formula contains three terms: the contribution of the internal symbol on the base manifold, expressed by an analog of the Atiyah–Singer integral, the contributions of the conormal symbols at infinity, described in terms of the \(\eta\) -invariant, and also the third term, which also depends on the conormal symbol. The result thus obtained generalizes the Fedosov–Schulze–Tarkhanov formula. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020091

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Abstract: We consider certain classes of operators generated by infinite band matrices (called band operators). They can be applied to the integration via the Lax pair formalism of nonlinear dynamical systems (e.g., Volterra type lattices) by using the inverse problem theory, in particular, by the inverse spectral problem method. A key role in this method is played by the moments of the Weyl matrix of a given band operator, which are used for unique reconstruction of the latter. These band operators have a special sparse structure, namely, they contain only two nonzero diagonals. We find a description of their sparsity in terms of these moments. PubDate: 2022-06-01 DOI: 10.1134/S1061920822020054

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Abstract: In many physical problems describing the propagation of waves, there are dispersion effects, both spatial and temporal. In these cases, the equations describing these problems and relating the temporal frequency and spatial momenta turn out to be inhomogeneous with respect to these variables. At the same time, these relations are even not polynomial. An example with time dispersion is given by Maxwell’s equations in a situation with rapidly changing electric and magnetic fields; then the second time derivative \(t\) is replaced by a function of this derivative (a pseudodifferential operator). In the paper, asymptotic formulas are constructed for the solution of the Cauchy problem with localized initial data for the equation \(g^2\bigl(-ih\frac{\partial}{\partial t}\bigr)u=-h^2\langle \nabla,\, c^2(x)\nabla\rangle u\) with variable speed and a small parameter characterizing fast oscillations of the propagating waves. One of the main considerations in use is that the constructive asymptotics for the solution of problems of this type are represented by functions given parametrically, and the corresponding parameters are the natural coordinates on the Lagrangian manifolds defining these asymptotics. Here, in contrast to problems with spatial dispersion, the corresponding Lagrangian manifold is defined in the phase space containing the time \(t\) and the corresponding conjugate momentum coordinate, the frequency \(\omega\) . PubDate: 2022-06-01 DOI: 10.1134/S1061920822020017

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Abstract: We prove that every locally bounded automorphism of a connected not necessarily linear Lie central extension of a connected perfect Lie group with discrete center is continuous if and only if it is continuous on the center. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010113

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Abstract: We consider the Cauchy problem for a parabolic equation with a Ё-Laplacian or a general second-order quasilinear equation with boundary conditions of the Bitsadze–Samarskii type. We prove that at least one generalized solution of such problem exists. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010125

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Abstract: The phase space of a gyrostat with a fixed point and a heavy top is the Lie–Poisson space \(\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3\) dual to the Lie algebra \(\textbf{e}(3)\) of the Euclidean group \(E(3)\) . One has three naturally distinguished Poisson submanifolds of \(\textbf{e}(3)^*\) : (i) the dense open submanifold \(\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*\) which consists of all \(4\) -dimensional symplectic leaves ( \(\vec{\Gamma}^2>0\) ); (ii) the \(5\) -dimensional Poisson submanifold of \(\mathbb{R}^3\times \dot{\mathbb{R}}^3\) defined by \(\vec{J}\cdot \vec{\Gamma} = \mu \vec{\Gamma} \) ; (iii) the \(5\) -dimensional Poisson submanifold of \(\mathbb{R}^3\times \dot{\mathbb{R}}^3\) defined by \(\vec{\Gamma}^2 = \nu^2\) , where \(\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}\) , \((\vec{J}, \vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*\) and \(\nu < 0 \) , \(\mu\) are some fixed real parameters. Using the \(U(2,2)\) -invariant symplectic structure of Penrose twistor space we find full and complete \(E(3)\) -equivariant symplectic realizations of these Poisson submanifolds which are \(8\) -dimensional for (i) and \(6\) -dimensional for (ii) and (iii). As a consequence of the above, Hamiltonian systems on \(\textbf{e}(3)^*\) lift to Hamiltonian systems on the above symplectic realizations. In this way, after lifting the integrable cases of a gyrostat with a fixed point and of a heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010095